You would need to first use iTunes to sync the app to your iPhone. Then use the app to back up your contacts. It will create a vCard (.csv file) as an attachment to an email that you send to yourself. Open the email and double-click on the attachment to import the contacts to your Mac.
The app will back up all your contacts so when you import them to your Mac you may end up with duplicates. If this occurs, open Address Book (Contacts if you're running Mountain Lion) and go to Card>Look For Duplicates, then merge any duplicates that are found. I'm not sure it will maintain the original contact groups but it should import the contacts belonging to the groups. i.e., you may have to re-create your groups, then drag and drop the appropriate contacts into the "non-iCloud" groups that you are trying to recover.